3. PHYSICAL PARAMETERS

In the early treatizes on this subject, the mean value assigned to it
will be found to be 40.000000. Later writers suspected that the
decimal point had been accidentally shifted, and that the proper value
was 400.00000.

Lewis Carroll, "The new method of evaluation as applied to
"
from Notes by an Oxford Chiel

Total intensity and polarization data can be used to extract
information about several physical parameters intrinsic to radio
sources and thereby help place constraints on some of the mechanisms
involved. Since the advent of computers and more recently of pocket
calculators, this "interpretation of the data" has become such an
automatic ritual that the many assumptions inherent in each step are
sometimes forgotten. We shall review here some of the arguments most
commonly used in deriving physical parameters from intensity and
polarized brightness distributions. In order to make the uncertainties
more explicit, the formulas will be expressed as much as possible in
terms of observed quantities. For block diagrams of the various steps
and assumptions that are made the reader is referred to
Miley (1976).

3.1.1 ENERGETICS The energetics of radio sources are
important not only because of
their relation to the source-production mechanism but because they
also play an important role in all considerations of how radio sources
are held together or confined. Total intensity distributions provide
information about the minimum energies involved. A good discussion of
source energetics is given by
Moffet (1975)
and in more detail by
Pacholcyzk (1970).
The minimum energy condition corresponds almost
(but not quite) to equipartition of energy between relativistic
particles and magnetic field.

For a region in a synchrotron radio source delineated by an ellipse
of angular diameters
x and
y in orthogonal
directions, we can write the minimum energy density as

(1)

where the corresponding magnetic field is

(2)

Here k is the ratio of energy in the heavy particles to that in the
electrons, is
the filling factor of the emitting regions, z is the redshift,
x and
y (arcsec)
correspond either to the source/component
sizes or to the equivalent beam widths, s (kiloparsec) is the path
length through the source in the line of sight,
is the angle between
the uniform magnetic field and the line of sight, F0
(Jy or Jy per beam) is the flux density or brightness of the region at
frequency 0
(GHz), 1 and
2 (GHz) are the
upper and lower cut off frequencies
presumed for the radio spectrum, and
is the spectral index
[F()
,
1 <
<
2].

Apart from the basic assumptions that the radiation is synchrotron
emission and that the radio and optical emission are redshifted by the
same amount there are several more mundane uncertainties inherent in
these formulas.

First, k is unknown and could have a value between 1 and 2000
(Pacholczyk 1970).
Indirect arguments have usually led to canonical
values of k = 1 or k = 100 (eg.
Moffet 1975).
k = 1 is clearly more
consistent with the use of the minimum energy condition. Note that a
difference of 100 in the assumed k results in an order-of-magnitude
difference in the minimum energy densities derived. Second, to obtain
s, the path through the source along the line of sight, one must make
some assumptions about the symmetry and distance of the
source. Frequently cylindrical symmetry is assumed, with s equal to
the width of the source in the plane of the sky. Third, the formulas
depend on the form of the source spectrum, but this dependence is
weak. For extended sources
-0.6, and
1 dominates over
2. Usually
1 = 0.01 GHz is
assumed. Fourth, the term is
unknown in individual cases. It arises because the emission measure
depends on the perpendicular component of the magnetic field and the
visible radiation is beamed from electrons moving towards us. Fifth,
and perhaps most irritating, is the strong dependence on the filling
factor. It is possible that on a scale much less than an arcsecond or
a kiloparsec the radiation is clumpy or filamentary. This would result
in much greater local energy densities and less total energy.

The minimum energy condition is particularly sensitive to angular
size. Because of the limited range of brightnesses that can be studied
with a particular instrument, values of Bme derived from
measurements
made with the same telescope usually do not differ by much more than a
factor of four or five. Typical values obtained for
Bme are ~ 10-6.5 for
cluster halos, 10-5.5 G for the diffuse lobes,
~ 10-5 G for the hot spots, and
~ 10-3 G for the tiny flat spectrum cores. Resultant
total source energies can reach 1060 ergs. However, it must be
stressed that there is little evidence that the minimum energy condition
is actually obeyed.

3.1.2 SOURCE CONFINEMENT In order to account for the
nonspherical shapes of radio sources the
particles and field must be prevented from dispersing at the
relativistic internal sound speed of c /
3. The pressure exerted
by the
relativistic gas in the radio source, u/3, must therefore be balanced
by some pressure, P, or by inertial drag. Several mechanisms have
been considered for confinement (see, for example,
Longair et al. 1973,
Pacholczyk 1977)
and in each case the pressure balance condition can
be used to calculate limits for the various parameters that
characterize the resisting forces.

Some possible confining mechanisms are listed in
Table 1 together
with the restrictions implied by pressure balance. Their applicability
to the various regions of radio sources will be considered in
Section 4. In real life several of these
processes may be occurring
simultaneously, and the detailed hydrodynamic interactions will
probably result in a considerably more complicated situation than can
be treated by simple static-pressure balance arguments.

The expression for the resultant magnetic field strength given by
Harris & Grindlay
(1979)
can be approximated to within about ± 10%
between -0.6 > >
-1.4 and rewritten as

(5)

where FR is the radio flux density (in Jy) at frequency
R (GHz),
Fx
is the X-ray flux (erg cm-2 Hz-1) at energy
Ex (keV),
is the spectral index
(FRR)
and z is the redshift.

The main problems in using this method to determine the magnetic
field strengths in extended radio sources arise from the required
X-ray surface brightness sensitivity and the difficulty of separating
possible inverse Compton X rays from the thermal X-ray
distribution. However, one can at least obtain an upper limit for the
inverse Compton contribution and hence a lower limit on the magnetic
field strengths. Until now this has been done using only a few
relatively low resolution X-ray measurements. For Centaurus ACooke et al. (1978)
obtained B
7 × 10-7 G and for Cygnus AFabbiano et al. (1979)
obtained B
1.6 × 10-6 G indicating that the
extended lobes
are not too far from equipartition. With high sensitivity X-ray
observations and resolution comparable to the radio maps, this method
promises to provide more information about magnetic fields in extended
radio sources. A beginning is now being made with the Einstein
Observatory, which, although providing an enormous improvement in
X-ray sensitivity and resolution, can detect extended inverse Compton
emission only from a few of the strongest sources.

3.1.4 AGES - IN SITU ACCELERATION A problem
frequently considered in discussions of source morphology
concerns the place where the observed synchrotron electrons are
accelerated. Does this occur entirely in the nucleus, mainly in the
hot spots, or is there an accelerating process at work throughout the
entire source?

For the acceleration to have taken place at an angular distance
(arcsec) from where the radiation is observed, the electron must have
traveled to its present position in time,

(6)

where D
(Mpc) is the "angular size" distance
(McVittie 1965),
v (km s-1) is the velocity at which electrons travel
from the nucleus to the position under study,
is the angle between this
direction and the
line of sight. For an isotropic distribution of pitch angles the
average age of a radiating synchrotron electron, tr, is
(van der Laan & Perola
1969,
De Young 1976)

(7)

where B (G) is the magnetic field strength in the source,
BR = 4 × 10-6(1 + z)2
G is the equivalent magnetic field strength of the microwave background.
* (GHz) is
the frequency above which an exponential drop
in flux will occur, and usually exceeds
0, the observing
frequency. A
necessary condition for acceleration to have occurred at a distance
is td = tr, i.e.

(8)

To apply this condition we need to have some information about v and
B. Of course we can always take v < c, but in
most cases there is
evidence that the velocities are considerably smaller. If we accept
the radio-trail hypothesis
(Section 2.2.1), for tailed sources
v is
the velocity of the parent galaxy. A reasonable upper limit to v of a
few thousand km s-1 is then given by the distribution of radial
velocities of galaxies in the cluster. Likewise, for the lobes of
double sources, the observed asymmetry has been used as a statistical
argument that the outward electron velocities are smaller than 0.2 c
(Sections 4.2 and
4.4). An additional possible upper limit to the
velocities is the generalized sound speed in the source (u /
)1/2
which in the equipartition case reduces to the Alfvén velocity
B/(4)1/2,
However, this is less certain since resonant damping by thermal
protons in the source might permit considerably higher velocities
(Holman et al. 1979).

There are two ways of specifying B. The first is to assume that the
minimum energy (~ equipartition) condition holds and to calculate B
from (2) or (3). The weaknesses of this approach are the lack of
evidence for equipartition and the uncertainties in the
formulas. (Note that minimum energy does not imply minimum B.) Also,
the assumption that the pitch angles
are isotropically
distributed may be invalid. In that case one must replace B by
B = B
sin, so
electrons flowing along the magnetic field lines will suffer smaller
energy losses and have prolonged lives
(Spangler 1979).

One can avoid most of these limitations by choosing
B = BR /
3, the
value that gives a maximum radiative lifetime
(van der Laan & Perola
1969).
The necessary condition for remote acceleration then becomes